Project acronym 1stProposal
Project An alternative development of analytic number theory and applications
Researcher (PI) ANDREW Granville
Host Institution (HI) UNIVERSITY COLLEGE LONDON
Call Details Advanced Grant (AdG), PE1, ERC-2014-ADG
Summary The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods.
Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject.
We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small.
We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration.
Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further.
Much of this is joint work with K Soundararajan of Stanford University.
Summary
The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods.
Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject.
We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small.
We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration.
Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further.
Much of this is joint work with K Soundararajan of Stanford University.
Max ERC Funding
2 011 742 €
Duration
Start date: 2015-08-01, End date: 2020-07-31
Project acronym ACB
Project The Analytic Conformal Bootstrap
Researcher (PI) Luis Fernando ALDAY
Host Institution (HI) THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
Call Details Advanced Grant (AdG), PE2, ERC-2017-ADG
Summary The aim of the present proposal is to establish a research team developing and exploiting innovative techniques to study conformal field theories (CFT) analytically. Our approach does not rely on a Lagrangian description but on symmetries and consistency conditions. As such it applies to any CFT, offering a unified framework to study generic CFTs analytically. The initial implementation of this program has already led to striking new results and insights for both Lagrangian and non-Lagrangian CFTs.
The overarching aims of my team will be: To develop an analytic bootstrap program for CFTs in general dimensions; to complement these techniques with more traditional methods and develop a systematic machinery to obtain analytic results for generic CFTs; and to use these results to gain new insights into the mathematical structure of the space of quantum field theories.
The proposal will bring together researchers from different areas. The objectives in brief are:
1) Develop an alternative to Feynman diagram computations for Lagrangian CFTs.
2) Develop a machinery to compute loops for QFT on AdS, with and without gravity.
3) Develop an analytic approach to non-perturbative N=4 SYM and other CFTs.
4) Determine the space of all CFTs.
5) Gain new insights into the mathematical structure of the space of quantum field theories.
The outputs of this proposal will include a new way of doing perturbative computations based on symmetries; a constructive derivation of the AdS/CFT duality; new analytic techniques to attack strongly coupled systems and invaluable new lessons about the space of CFTs and QFTs.
Success in this research will lead to a completely new, unified way to view and solve CFTs, with a huge impact on several branches of physics and mathematics.
Summary
The aim of the present proposal is to establish a research team developing and exploiting innovative techniques to study conformal field theories (CFT) analytically. Our approach does not rely on a Lagrangian description but on symmetries and consistency conditions. As such it applies to any CFT, offering a unified framework to study generic CFTs analytically. The initial implementation of this program has already led to striking new results and insights for both Lagrangian and non-Lagrangian CFTs.
The overarching aims of my team will be: To develop an analytic bootstrap program for CFTs in general dimensions; to complement these techniques with more traditional methods and develop a systematic machinery to obtain analytic results for generic CFTs; and to use these results to gain new insights into the mathematical structure of the space of quantum field theories.
The proposal will bring together researchers from different areas. The objectives in brief are:
1) Develop an alternative to Feynman diagram computations for Lagrangian CFTs.
2) Develop a machinery to compute loops for QFT on AdS, with and without gravity.
3) Develop an analytic approach to non-perturbative N=4 SYM and other CFTs.
4) Determine the space of all CFTs.
5) Gain new insights into the mathematical structure of the space of quantum field theories.
The outputs of this proposal will include a new way of doing perturbative computations based on symmetries; a constructive derivation of the AdS/CFT duality; new analytic techniques to attack strongly coupled systems and invaluable new lessons about the space of CFTs and QFTs.
Success in this research will lead to a completely new, unified way to view and solve CFTs, with a huge impact on several branches of physics and mathematics.
Max ERC Funding
2 171 483 €
Duration
Start date: 2018-12-01, End date: 2023-11-30
Project acronym AMSTAT
Project Problems at the Applied Mathematics-Statistics Interface
Researcher (PI) Andrew Stuart
Host Institution (HI) THE UNIVERSITY OF WARWICK
Call Details Advanced Grant (AdG), PE1, ERC-2008-AdG
Summary Applied mathematics is concerned with developing models with predictive capability, and with probing those models to obtain qualitative and quantitative insight into the phenomena being modelled. Statistics is data-driven and is aimed at the development of methodologies to optimize the information derived from data. The increasing complexity of phenomena that scientists and engineers wish to model, together with our increased ability to gather, store and interrogate data, mean that the subjects of applied mathematics and statistics are increasingly required to work in conjunction. This research proposal is concerned with a research program at the interface between these two disciplines, aimed at problems in differential equations where profusion of data and the sophisticated model combine to produce the mathematical problem of obtaining information from a probability measure on function space. Applications are far-reaching and include the atmospheric sciences, geophysics, chemistry, econometrics and signal processing. The objectives of the research are: (i) to create the systematic foundations for a range of problems at the applied mathematics and statistics interface which share the common mathematical structure underpinning the range of applications described above; (ii) to exploit this common mathematical structure to design effecient algorithms to sample probability measures on function space; (iii) to apply these algorithms to attack a range of significant problems arising in molecular dynamics and in the atmospheric sciences.
Summary
Applied mathematics is concerned with developing models with predictive capability, and with probing those models to obtain qualitative and quantitative insight into the phenomena being modelled. Statistics is data-driven and is aimed at the development of methodologies to optimize the information derived from data. The increasing complexity of phenomena that scientists and engineers wish to model, together with our increased ability to gather, store and interrogate data, mean that the subjects of applied mathematics and statistics are increasingly required to work in conjunction. This research proposal is concerned with a research program at the interface between these two disciplines, aimed at problems in differential equations where profusion of data and the sophisticated model combine to produce the mathematical problem of obtaining information from a probability measure on function space. Applications are far-reaching and include the atmospheric sciences, geophysics, chemistry, econometrics and signal processing. The objectives of the research are: (i) to create the systematic foundations for a range of problems at the applied mathematics and statistics interface which share the common mathematical structure underpinning the range of applications described above; (ii) to exploit this common mathematical structure to design effecient algorithms to sample probability measures on function space; (iii) to apply these algorithms to attack a range of significant problems arising in molecular dynamics and in the atmospheric sciences.
Max ERC Funding
1 693 501 €
Duration
Start date: 2008-12-01, End date: 2014-11-30
Project acronym ARITHQUANTUMCHAOS
Project Arithmetic and Quantum Chaos
Researcher (PI) Zeev Rudnick
Host Institution (HI) TEL AVIV UNIVERSITY
Call Details Advanced Grant (AdG), PE1, ERC-2012-ADG_20120216
Summary Quantum Chaos is an emerging discipline which is crossing over from Physics into Pure Mathematics. The recent crossover is driven in part by a connection with Number Theory. This project explores several aspects of this interrelationship and is composed of a number of sub-projects. The sub-projects deal with: statistics of energy levels and wave functions of pseudo-integrable systems, a hitherto unexplored subject in the mathematical community which is not well understood in the physics community; with statistics of zeros of zeta functions over function fields, a purely number theoretic topic which is linked to the subproject on Quantum Chaos through the mysterious connections to Random Matrix Theory and an analogy between energy levels and zeta zeros; and with spatial statistics in arithmetic.
Summary
Quantum Chaos is an emerging discipline which is crossing over from Physics into Pure Mathematics. The recent crossover is driven in part by a connection with Number Theory. This project explores several aspects of this interrelationship and is composed of a number of sub-projects. The sub-projects deal with: statistics of energy levels and wave functions of pseudo-integrable systems, a hitherto unexplored subject in the mathematical community which is not well understood in the physics community; with statistics of zeros of zeta functions over function fields, a purely number theoretic topic which is linked to the subproject on Quantum Chaos through the mysterious connections to Random Matrix Theory and an analogy between energy levels and zeta zeros; and with spatial statistics in arithmetic.
Max ERC Funding
1 714 000 €
Duration
Start date: 2013-02-01, End date: 2019-01-31
Project acronym ASTEX
Project Attosecond Science by Transmission and Emission of X-rays
Researcher (PI) Jonathan Philip Marangos
Host Institution (HI) IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
Call Details Advanced Grant (AdG), PE2, ERC-2011-ADG_20110209
Summary "This is a programme of advanced research with potential for high scientific impact and applications to areas of great strategic importance such as renewable energy and biomolecular technology. The aim is to develop and apply a combination of cutting-edge tools to observe and understand dynamics in molecules and condensed phase matter with attosecond temporal and nanometre spatial resolutions. The programme, will exploit two new types of measurements that my group have already begun to develop: high harmonic generation (HHG) spectroscopy and attosecond absorption pump-probe spectroscopy, and will apply them to the measurement of attosecond electron dynamics in large molecules and the condensed phase. These methods rely upon the emission and transmission of soft X-ray attosecond fields that make accessible measurement not only of larger molecules in the gas phase but also thin (micron to nanometre) samples in the condensed phase. This is a research project that will open new frontiers both experimentally and theoretically. The challenge of this research is high and will be met by a concerted programme that is well matched to my teams experimental and theoretical expertise in attosecond physics, ultrafast intense-field science, soft X-ray techniques and advanced techniques for creating gaseous and condensed phase samples."
Summary
"This is a programme of advanced research with potential for high scientific impact and applications to areas of great strategic importance such as renewable energy and biomolecular technology. The aim is to develop and apply a combination of cutting-edge tools to observe and understand dynamics in molecules and condensed phase matter with attosecond temporal and nanometre spatial resolutions. The programme, will exploit two new types of measurements that my group have already begun to develop: high harmonic generation (HHG) spectroscopy and attosecond absorption pump-probe spectroscopy, and will apply them to the measurement of attosecond electron dynamics in large molecules and the condensed phase. These methods rely upon the emission and transmission of soft X-ray attosecond fields that make accessible measurement not only of larger molecules in the gas phase but also thin (micron to nanometre) samples in the condensed phase. This is a research project that will open new frontiers both experimentally and theoretically. The challenge of this research is high and will be met by a concerted programme that is well matched to my teams experimental and theoretical expertise in attosecond physics, ultrafast intense-field science, soft X-ray techniques and advanced techniques for creating gaseous and condensed phase samples."
Max ERC Funding
2 344 390 €
Duration
Start date: 2012-04-01, End date: 2017-03-31
Project acronym CHROMIUM
Project CHROMIUM
Researcher (PI) Jennifer THOMAS
Host Institution (HI) UNIVERSITY COLLEGE LONDON
Call Details Advanced Grant (AdG), PE2, ERC-2015-AdG
Summary Why the Universe is void of anti-matter is one of the remaining Big Questions in Science.One explanation is provided within the Standard Model by violation of Charge Parity (CP) symmetry, producing differences between the behavior of particles and their anti-particles.CP violation in the neutrino sector could allow a mechanism by which the matter-anti matter asymmetry arose.The objective of this proposal is to enable a step change in our sensitivity to CP violation in the neutrino sector. I have pioneered the concepts and led the deployment of a small prototype using a novel approach which could eventually lead to the construction of a revolutionary Mega-ton scale Water Cherenkov (WC) neutrino detector.The goal of my research program is to demonstrate the feasibility of this approach via the construction of an intermediate sized prototype with an expandable fiducial mass of up to 10-20kt. It will use a low-cost and lightweight structure, filled with purified water and submerged for mechanical strength and cosmic ray shielding in a 60m deep flooded mine pit in the path of Fermilab’s NuMI neutrino beam in N. Minnesota.The European contribution to this experiment will be profound and definitive.Applying the idea of fast timing and good position resolution of small photodetectors, already pioneered in Europe, in place of large-area photodetector, we will revolutionize WC design.The game-changing nature of this philosophy will be demonstrated via the proof of the detector construction and the observation of electron neutrino events form the NuMI beam.The successful completion of this R&D program will demonstrate a factor of up to 100 decrease in cost compared to conventional detectors and the proof that precision neutrino measurements could be made inside a few years rather than the presently needed decades.
The project describes a five year program of work amounting to a total funding request of €3.5M, including an extra €1M of equipment funds.
Summary
Why the Universe is void of anti-matter is one of the remaining Big Questions in Science.One explanation is provided within the Standard Model by violation of Charge Parity (CP) symmetry, producing differences between the behavior of particles and their anti-particles.CP violation in the neutrino sector could allow a mechanism by which the matter-anti matter asymmetry arose.The objective of this proposal is to enable a step change in our sensitivity to CP violation in the neutrino sector. I have pioneered the concepts and led the deployment of a small prototype using a novel approach which could eventually lead to the construction of a revolutionary Mega-ton scale Water Cherenkov (WC) neutrino detector.The goal of my research program is to demonstrate the feasibility of this approach via the construction of an intermediate sized prototype with an expandable fiducial mass of up to 10-20kt. It will use a low-cost and lightweight structure, filled with purified water and submerged for mechanical strength and cosmic ray shielding in a 60m deep flooded mine pit in the path of Fermilab’s NuMI neutrino beam in N. Minnesota.The European contribution to this experiment will be profound and definitive.Applying the idea of fast timing and good position resolution of small photodetectors, already pioneered in Europe, in place of large-area photodetector, we will revolutionize WC design.The game-changing nature of this philosophy will be demonstrated via the proof of the detector construction and the observation of electron neutrino events form the NuMI beam.The successful completion of this R&D program will demonstrate a factor of up to 100 decrease in cost compared to conventional detectors and the proof that precision neutrino measurements could be made inside a few years rather than the presently needed decades.
The project describes a five year program of work amounting to a total funding request of €3.5M, including an extra €1M of equipment funds.
Max ERC Funding
3 500 000 €
Duration
Start date: 2016-10-01, End date: 2021-09-30
Project acronym COIMBRA
Project Combinatorial methods in noncommutative ring theory
Researcher (PI) Agata Smoktunowicz
Host Institution (HI) THE UNIVERSITY OF EDINBURGH
Call Details Advanced Grant (AdG), PE1, ERC-2012-ADG_20120216
Summary As noted by T Y Lam in his book, A first course in noncommutative rings, noncommutative ring theory is a fertile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential operators), noncommutative algebraic geometry (graded domains), arithmetic (orders, Brauer groups), universal algebra (co-homology of rings, projective modules) and quantum physics (quantum matrices). As such, noncommutative ring theory is an area which has the potential to produce developments in many areas and in an efficient manner. The main aim of the project is to develop methods which could be applicable not only in ring theory but also in other areas, and then apply them to solve several important open questions in mathematics. The Principal Investigator, along with two PhD students and two post doctorates, propose to: study basic open questions on infinite dimensional associative noncommutative algebras; pool their expertise so as to tackle problems from a number of related areas of mathematics using noncommutative ring theory, and develop new approaches to existing problems that will benefit future researchers. A part of our methodology would be to first improve (in some cases) Bergman's Diamond Lemma, and then apply it to several open problems. The Diamond Lemma gives bases for the algebras defined by given sets of relations. In general, it is very difficult to determine if the algebra given by a concrete set of relations is non-trivial or infinite dimensional. Our approach is to introduce smaller rings, which we will call platinum rings. The next step would then be to apply the Diamond Lemma to the platinum ring instead of the original rings. Such results would have many applications in group theory, noncommutative projective geometry, nonassociative algebras and no doubt other areas as well.
Summary
As noted by T Y Lam in his book, A first course in noncommutative rings, noncommutative ring theory is a fertile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential operators), noncommutative algebraic geometry (graded domains), arithmetic (orders, Brauer groups), universal algebra (co-homology of rings, projective modules) and quantum physics (quantum matrices). As such, noncommutative ring theory is an area which has the potential to produce developments in many areas and in an efficient manner. The main aim of the project is to develop methods which could be applicable not only in ring theory but also in other areas, and then apply them to solve several important open questions in mathematics. The Principal Investigator, along with two PhD students and two post doctorates, propose to: study basic open questions on infinite dimensional associative noncommutative algebras; pool their expertise so as to tackle problems from a number of related areas of mathematics using noncommutative ring theory, and develop new approaches to existing problems that will benefit future researchers. A part of our methodology would be to first improve (in some cases) Bergman's Diamond Lemma, and then apply it to several open problems. The Diamond Lemma gives bases for the algebras defined by given sets of relations. In general, it is very difficult to determine if the algebra given by a concrete set of relations is non-trivial or infinite dimensional. Our approach is to introduce smaller rings, which we will call platinum rings. The next step would then be to apply the Diamond Lemma to the platinum ring instead of the original rings. Such results would have many applications in group theory, noncommutative projective geometry, nonassociative algebras and no doubt other areas as well.
Max ERC Funding
1 406 551 €
Duration
Start date: 2013-06-01, End date: 2018-05-31
Project acronym DEPENDENTCLASSES
Project Model theory and its applications: dependent classes
Researcher (PI) Saharon Shelah
Host Institution (HI) THE HEBREW UNIVERSITY OF JERUSALEM
Call Details Advanced Grant (AdG), PE1, ERC-2013-ADG
Summary Model theory deals with general classes of structures (called models).
Specific examples of such classes are: the class of rings or the class of
algebraically closed fields.
It turns out that counting the so-called complete types over models in the
class has an important role in the development of model theory in general and
stability theory in particular.
Stable classes are those with relatively few complete types (over structures
from the class); understanding stable classes has been central in model theory
and its applications.
Recently, I have proved a new dichotomy among the unstable classes:
Instead of counting all the complete types, they are counted up to conjugacy.
Classes which have few types up to conjugacy are proved to be so-called
``dependent'' classes (which have also been called NIP classes).
I have developed (under reasonable restrictions) a ``recounting theorem'',
parallel to the basic theorems of stability theory.
I have started to develop some of the basic properties of this new approach.
The goal of the current project is to develop systematically the theory of
dependent classes. The above mentioned results give strong indication that this
new theory can be eventually as useful as the (by now the classical) stability
theory. In particular, it covers many well known classes which stability theory
cannot treat.
Summary
Model theory deals with general classes of structures (called models).
Specific examples of such classes are: the class of rings or the class of
algebraically closed fields.
It turns out that counting the so-called complete types over models in the
class has an important role in the development of model theory in general and
stability theory in particular.
Stable classes are those with relatively few complete types (over structures
from the class); understanding stable classes has been central in model theory
and its applications.
Recently, I have proved a new dichotomy among the unstable classes:
Instead of counting all the complete types, they are counted up to conjugacy.
Classes which have few types up to conjugacy are proved to be so-called
``dependent'' classes (which have also been called NIP classes).
I have developed (under reasonable restrictions) a ``recounting theorem'',
parallel to the basic theorems of stability theory.
I have started to develop some of the basic properties of this new approach.
The goal of the current project is to develop systematically the theory of
dependent classes. The above mentioned results give strong indication that this
new theory can be eventually as useful as the (by now the classical) stability
theory. In particular, it covers many well known classes which stability theory
cannot treat.
Max ERC Funding
1 748 000 €
Duration
Start date: 2014-03-01, End date: 2019-02-28
Project acronym DG-PESP-CS
Project Deterministic Generation of Polarization Entangled single Photons Cluster States
Researcher (PI) David Gershoni
Host Institution (HI) TECHNION - ISRAEL INSTITUTE OF TECHNOLOGY
Call Details Advanced Grant (AdG), PE2, ERC-2015-AdG
Summary Measurement based quantum computing is one of the most fault-tolerant architectures proposed for quantum information processing. It opens the possibility of performing quantum computing tasks using linear optical systems. An efficient route for measurement based quantum computing utilizes highly entangled states of photons, called cluster states. Propagation and processing quantum information is made possible this way using only single qubit measurements. It is highly resilient to qubit losses. In addition, single qubit measurements of polarization qubits is easily performed with high fidelity using standard optical tools. These features make photonic clusters excellent platforms for quantum information processing.
Constructing photonic cluster states, however, is a formidable challenge, attracting vast amounts of research efforts. While in principle it is possible to build up cluster states using interferometry, such a method is of a probabilistic nature and entails a large overhead of resources. The use of entangled photon pairs reduces this overhead by a small factor only.
We outline a novel route for constructing a deterministic source of photonic cluster states using a device based on semiconductor quantum dot. Our proposal follows a suggestion by Lindner and Rudolph. We use repeated optical excitations of a long lived coherent spin confined in a single semiconductor quantum dot and demonstrate for the first time practical realization of their proposal. Our preliminary demonstration presents a breakthrough in quantum technology since deterministic source of photonic cluster, reduces the resources needed quantum information processing. It may have revolutionary prospects for technological applications as well as to our fundamental understanding of quantum systems.
We propose to capitalize on this recent breakthrough and concentrate on R&D which will further advance this forefront field of science and technology by utilizing the horizons that it opens.
Summary
Measurement based quantum computing is one of the most fault-tolerant architectures proposed for quantum information processing. It opens the possibility of performing quantum computing tasks using linear optical systems. An efficient route for measurement based quantum computing utilizes highly entangled states of photons, called cluster states. Propagation and processing quantum information is made possible this way using only single qubit measurements. It is highly resilient to qubit losses. In addition, single qubit measurements of polarization qubits is easily performed with high fidelity using standard optical tools. These features make photonic clusters excellent platforms for quantum information processing.
Constructing photonic cluster states, however, is a formidable challenge, attracting vast amounts of research efforts. While in principle it is possible to build up cluster states using interferometry, such a method is of a probabilistic nature and entails a large overhead of resources. The use of entangled photon pairs reduces this overhead by a small factor only.
We outline a novel route for constructing a deterministic source of photonic cluster states using a device based on semiconductor quantum dot. Our proposal follows a suggestion by Lindner and Rudolph. We use repeated optical excitations of a long lived coherent spin confined in a single semiconductor quantum dot and demonstrate for the first time practical realization of their proposal. Our preliminary demonstration presents a breakthrough in quantum technology since deterministic source of photonic cluster, reduces the resources needed quantum information processing. It may have revolutionary prospects for technological applications as well as to our fundamental understanding of quantum systems.
We propose to capitalize on this recent breakthrough and concentrate on R&D which will further advance this forefront field of science and technology by utilizing the horizons that it opens.
Max ERC Funding
2 502 974 €
Duration
Start date: 2016-06-01, End date: 2021-05-31
Project acronym DIFFERENTIALGEOMETR
Project Geometric analysis, complex geometry and gauge theory
Researcher (PI) Simon Kirwan Donaldson
Host Institution (HI) IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
Call Details Advanced Grant (AdG), PE1, ERC-2009-AdG
Summary The proposal is for work in Geometric Analysis aimed at two different problems. One is to establish necessary and sufficient conditions for the existence of extremal metrics on complex algebraic manifolds. These metrics are characterised by conditions on their curvature tensor a paradigm being the Riemannian version of the Einstein equation of General Relativity The standard conjecture is that the right condition should be the stability of the manifold, a condition defined entirely in the language of algebraic geometry. But there are very few cases where this conjecture has been verified. The problem comes down to proving the existence of a solution to highly nonlinear partial differential equation. The aim is to advance this theory by a detailed study of interesting but more amenable cases, for example where there is a large symmetry group. The second problem is to develop new invariants and structures associated to a particular class of manifolds of dimension 6 and 7 (with holonomy SU(3) and G2). These would be derived from the solutions of versions of the Yang-Mills equation over the manifolds, in a similar manner to familiar theories in 3 and 4 dimensions. In higher dimensions there are fundamental new difficulties to overcome to set up a theory rigorously and the main point of this part of the proposal is to attack these. It is likely that the new structures, if they do exist, will have interesting connections to other developments in this general area, involving string theory and algebraic geometry.
Summary
The proposal is for work in Geometric Analysis aimed at two different problems. One is to establish necessary and sufficient conditions for the existence of extremal metrics on complex algebraic manifolds. These metrics are characterised by conditions on their curvature tensor a paradigm being the Riemannian version of the Einstein equation of General Relativity The standard conjecture is that the right condition should be the stability of the manifold, a condition defined entirely in the language of algebraic geometry. But there are very few cases where this conjecture has been verified. The problem comes down to proving the existence of a solution to highly nonlinear partial differential equation. The aim is to advance this theory by a detailed study of interesting but more amenable cases, for example where there is a large symmetry group. The second problem is to develop new invariants and structures associated to a particular class of manifolds of dimension 6 and 7 (with holonomy SU(3) and G2). These would be derived from the solutions of versions of the Yang-Mills equation over the manifolds, in a similar manner to familiar theories in 3 and 4 dimensions. In higher dimensions there are fundamental new difficulties to overcome to set up a theory rigorously and the main point of this part of the proposal is to attack these. It is likely that the new structures, if they do exist, will have interesting connections to other developments in this general area, involving string theory and algebraic geometry.
Max ERC Funding
1 501 361 €
Duration
Start date: 2010-04-01, End date: 2015-03-31